Good formal matrix rings over residue class rings

Let $p$ be a prime number, $m, n$ be natural and $m\geqslant n>0$. Let the formal matrix ring $\begin{pmatrix} \mathbf{Z}/p^m\mathbf{Z} & \mathbf{Z}/p^n\mathbf{Z}\\ \mathbf{Z}/p^n\mathbf{Z} & \mathbf{Z}/p^n\mathbf{Z} \end{pmatrix}$ be isomorphic to the endomorphism ring $E((\mathbf{Z}/p^m\mathbf{Z})\oplus (\mathbf{Z}/p^n\mathbf{Z}))$, may be of interest in data encryption. We will show that the ring $E((\mathbf{Z}/p^m\mathbf{Z})\oplus (\mathbf{Z}/p^n\mathbf{Z}))$, $m\geqslant n$, is $2$-good and $2$-nil-good for $p > 2$ and not good for $p = 2$ and $m > n$.